Compute electric flux through a planar surface in a uniform electric field. Visualize field lines, surface normal, and angle dependency.
Electric flux ΦE quantifies the number of electric field lines passing through a given surface. For a uniform electric field E and a flat surface of area A, the flux is defined as Φ = E · A = E A cosθ, where θ is the angle between the electric field vector and the surface normal vector. This scalar quantity (measured in N·m²/C or V·m) can be positive, negative, or zero depending on orientation.
ΦE = ∮ E · dA → Φ = E A cosθ (uniform field, planar surface)
Gauss's law, one of Maxwell's equations, states that the total electric flux through a closed surface equals the enclosed charge divided by ε₀: Φtotal = Qenc / ε₀. This principle is fundamental to electrostatics, enabling calculation of fields for symmetric charge distributions. Our calculator focuses on the planar surface case, essential for building intuition before tackling closed surfaces.
The diagram dynamically updates: the gray rectangle represents a planar surface, the orange arrow is the unit normal vector, and the green parallel lines (with arrowheads) show the electric field direction relative to the normal. Adjust angle θ and see how the projected area changes, altering the number of field lines crossing the surface. This visual feedback reinforces the cosθ dependence.
| Scenario | E (N/C) | A (m²) | θ (deg) | Flux (N·m²/C) | Interpretation |
|---|---|---|---|---|---|
| Perpendicular field | 10 | 2.0 | 0 | 20.0 | Maximum positive flux |
| 45° incidence | 10 | 2.0 | 45 | 14.14 | Reduced by √2/2 |
| Grazing (parallel) | 10 | 2.0 | 90 | 0.0 | No field lines cross |
| Opposite normal | 10 | 2.0 | 180 | -20.0 | Flux is negative |
| Sphere equivalent demo | 5 | 1.0 | 0 | 5.0 | Uniform radial approximation |
Engineers calculate electric flux analogously for sunlight (intensity vector). For a solar panel of area 2.5 m², if the Sun's rays are at θ=30° from normal, the effective area (A cosθ) drops, reducing power. Similarly, in electrostatics, flux represents field line density. This tool's core principle helps design sensors, Faraday cages, and capacitance geometries.
For non‑uniform electric fields or curved surfaces, the general definition uses a surface integral: Φ = ∫ E·dA. Our calculator provides the fundamental uniform‑field case, a building block for understanding flux through any surface via infinitesimal patches. Gauss's law elegantly shows that for any closed surface, flux depends only on internal charge, not shape. Use this tool to verify that tilting a surface doesn't change flux if the net enclosed charge remains same, but here we focus on open surfaces.